Critical Sobolev exponent for degenerate elliptic operators (Q1267359)

The author investigates the boundary value problem \[ u_{xx} +x^{2k} u_{yy} + g(u)=0 \quad \text{in } \Omega, \quad u|_{\partial \Omega}=0, \] where \(g\in C(\mathbb{R})\), \(g(0) =0\), and \(\Omega\) is a starshaped domain in \(\mathbb{R}^2\). Using embedding theorems for suitable Sobolev energy spaces and a generalized Pokhozhaev identity, the author proves some existence and non-existence results under various technical assumptions on \(g(u)\).